Periodic Orbit Approach to the Quantum‐Kramers‐Rate

Hänggi, Peter; Hontscha, Waldemar

doi:10.1002/bbpc.19910950327

Cited by 31 publications

(12 citation statements)

References 35 publications

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“…70 The trajectories also describe dynamical recrossing effects to go beyond the TST approximation as in classical rate theory. Like RPMD, instanton theory can also describe the effects of Kramers' strong-friction regime 71 by coupling to bath modes in the same way as classical TST. 72 However, as it is derived in terms only of imaginary-time trajectories, instanton theory effectively ignores the dynamics of the system outside the barrier region and cannot therefore describe the transition rate of a double-well potential in Kramers' weak-friction regime.…”

Section: Connection To Other Path-integral Approachesmentioning

confidence: 99%

Perspective: Ring-polymer instanton theory

Richardson

2018

The Journal of Chemical Physics

124

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Since the earliest explorations of quantum mechanics, it has been a topic of great interest that quantum tunneling allows particles to penetrate classically insurmountable barriers. Instanton theory provides a simple description of these processes in terms of dominant tunneling pathways. Using a ring-polymer discretization, an efficient computational method is obtained for applying this theory to compute reaction rates and tunneling splittings in molecular systems. Unlike other quantum-dynamics approaches, the method scales well with the number of degrees of freedom, and for many polyatomic systems, the method may provide the most accurate predictions which can be practically computed. Instanton theory thus has the capability to produce useful data for many fields of low-temperature chemistry including spectroscopy, atmospheric and astrochemistry, as well as surface science. There is however still room for improvement in the efficiency of the numerical algorithms, and new theories are under development for describing tunneling in nonadiabatic transitions.

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Section: Connection To Other Path-integral Approachesmentioning

confidence: 99%

Perspective: Ring-polymer instanton theory

Richardson

2018

The Journal of Chemical Physics

124

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“…For this limit, we used the semiclassical approach of Miller. 2,18 The bounce trajectory for the adiabatic case can be found quite efficiently using a hybrid monodromy-shoot implementation of the Newton-Raphson technique. 19 For small ⌬, the Golden Rule expression from perturbation theory should hold.…”

Section: A Single Oscillator Spin-boson Modelmentioning

confidence: 99%

The semiclassical calculation of nonadiabatic tunneling rates

Schwieters¹,

Voth²

1998

The Journal of Chemical Physics

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Nonadiabatic instanton calculation of multistate electron transfer reaction rate: Interference effects in three and four states systemsIn this paper semiclassical low-temperature rate theory is extended to treat nonadiabatic transitions which are typically important in electron transfer reactions. This theory is appropriate for arbitrary coupling strength between electronic states. As in adiabatic semiclassical tunneling theory, it is found that the leading order contribution to the tunneling rate is due to periodic orbits which exist in the barrier region of configuration space between reactant and product. In the current case, these orbits move on effective potentials generated from upside-down ͑nuclear͒ potentials of the coupled electronic states. A stable method of finding these mixed quantum/classical ''trajectories'' is developed using a Newton-Raphson method. Examples employing model systems demonstrate that the current nonadiabatic theory well-reproduces the known adiabatic and Golden Rule limits and that the theory can indeed be applied to systems with more than one degree of freedom.

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“…It should be emphasized that the given arguments do not represent a proof of the relation between the decay rate and the imaginary part of the free energy in the dissipative case. However, there exist independent methods employing periodic orbit [21,24] or real-time path integral techniques [25] which lead to the same results for the decay rates. For details we refer the reader to the literature.…”

Section: Imaginary Part Of the Free Energymentioning

confidence: 99%

Dissipative Quantum Systems

Dattagupta¹

2008

A Paradigm Called Magnetism

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Dissipation is a ubiquitous phenomenon in real physical systems. Its nature is made clear by considering the damped harmonic oscillator, a paradigm for dissipative systems in the classical as well as the quantum regime. After starting at a nonequilibrium position, the system will perform damped oscillations and end up in the equilibrium position. Looking closely, one will notice that even in equilibrium the oscillator coordinate fluctuates. This effect is related to the Brownian motion of a free particle. Damping and fluctuations are both caused by the coupling of the harmonic oscillator to other degrees of freedom. A pendulum's motion is damped because of collisions with molecules in the air during the oscillations. For the same reason the pendulum will fluctuate around its equilibrium position. The identical origin of both effects manifests itself in the fluctuation-dissipation theorem which will be discussed later in this chapter.Another system where the coupling to other degrees of freedom plays a prominent role, is the decay of a metastable state. At not too low temperatures a state in a potential minimum, which is separated by a barrier from an energetically lower region, may decay by thermal activation. Here, the other degrees of freedom, which are also collectively called heat bath or reservoir, provide the system with the necessary energy to surmount the barrier. At low enough temperatures, the decay of the metastable state will be dominated by quantum tunneling through the barrier, and one might ask how the other degrees of freedom influence the tunneling rate.In this chapter, we will specifically consider the damped harmonic oscillator, the damped free particle, and the decay of a metastable state to illustrate the techniques introduced to describe dissipation in quantum mechanics. Of course, there are other interesting problems like the damped motion in a periodic potential and the dissipative two-level system [1]. For a more complete discussion of many aspects of quantum dissipation, we refer the reader e.g. to the textbook by Weiss [2] which may also serve as a guide to the literature.A rather recent area of research activities is concerned with the interplay between quantum dissipation and chaos which is discussed in Chapter 6. In this context, techniques for the treatment of driven systems discussed in Chapter 5 are required.

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Periodic Orbit Approach to the Quantum‐Kramers‐Rate

Cited by 31 publications

References 35 publications

Perspective: Ring-polymer instanton theory

Perspective: Ring-polymer instanton theory

The semiclassical calculation of nonadiabatic tunneling rates

Dissipative Quantum Systems

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